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The short answer is that: Maximizing the expected logarithm leads to more wealth almost surely in the long run. In contrast, maximizing expected return can easily lead to going broke almost surely in the long run! Maximizing expected return results in betting everything on your highest expected return investment. Repeatedly doing that over time typically ...


11

Maximizing $E[\log(G)]$ which corresponds to a concave utility function is a subtle way of incorporating risk aversion in the utility. Maximizing $E[G]$ is basically saying that you have linear utility which corresponds to infinite risk appetite because as soon as you have positive expectation you are willing to bet as much capital as possible no matter the ...


7

This problem can be expressed as the original Merton's portfolio problem. Consider wealth process defined by SDE $$ d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac { X _ { t } \left( 1 - \alpha _ { t } \right) } { S _ { t } ^ { 0 } } d S _ { t } ^ { 0 } $$ where $\alpha_t$ is proportion of the investment in the risky ...


6

The original paper was concerned with optimizing the long run geometric return. In fact, he does not explicitly optimize either $\mathbb{E}(G)$ or $\mathbb{E}(\log(G))$. He also assumes the probabilities are known. The expectation is implicit in his assumption that $$G=\lim_{N\to\infty}\frac{1}{N}\log\left(\frac{V_N}{V_0}\right).$$ He notes that $$V_N=(...


5

This is indeed a very good question! There were (and still are) very hefty debates, where even academic champions like Paul Samuelson were involved! A very good starting point to get some main arguments is the following chapter 4 from the book "Fortunes Formula" by William Poundstone: https://books.google.de/books?id=xz4y3u-qM04C&lpg=PA179&dq=...


3

"Money management" is the art or business of managing money on behalf of an investor (individual or institutions such as pension fund, college endowment, etc.). Money managers usually organize their work by making decisions in a hierarchical fashion on 3 different levels: Strategic Asset Allocation (highest level, few decisions - infrequently ...


3

Hint: you are looking for weights w1,w2,⋯,w15 such that the linear combination of the 15 stocks daily returns is maximally correlated with the S&P500 index returns published by S&P. There are method(s) in statistics that can find these weights. Some of these techniques restrict the weights to be positive and some do not. There is plenty of historical ...


2

When it comes to Kelly, I've always liked the explanation at http://www.financialwisdomforum.org/gummy-stuff/kelly-ratio.htm. At this link there are three versions of Kelly explained, if you don't mind the "chatty" style. The third version brings in the standard deviation of wins and losses, which I think is very useful.


2

According to Skiena (link page 21) the Kelly fraction in the case of wins all equal to W and losses all equal to L is: $$f=\frac{pW-qL}{WL}=p/L-q/W$$ where $q=1-p$ and $p$ is the probability of a win. When the wins and losses are random, with average $W$ and $L$ respectively, I am not sure this formula is completely justified. But it might be a good ...


2

Maginn et al. (2007) suggest one of two approaches: optimization and stratified sampling. Optimization. First, you select some factor model which (you believe) best captures major risk factors in your universe. Then you select portfolio of 15 stocks so that this resulting portfolio would minimize expected tracking risk of original portfolio (S&P500 in ...


1

Two thoughts that I'm interested in at the moment... A Deep Hedging-stype approach to risk management (eg. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3355706) Fundamentally all of derivative pricing quant finance is Model-Driven. Why? Probably mostly because it's easier. However, we now have maybe 30 years of high frequency tick data available ...


1

I've recently had to do quite a bit of work on position sizing. Leonard C MacLean, Edward O Thorp, and William T Ziemba have written an incredible amount of literature on this. The following text book encompasses an incredibly deep study of the topic on position sizing, different utility functions and so on. From what I can tell the two broad branches of ...


1

Kelly maximises the geometric return in each period. I believe this is equivalent to maximising expectation of log wealth in the next period. If you maximise the arithmetic return, your expected wealth will be higher. However the actual return you will experience is -100% i.e. you will go bust with probability 1. If you maximise the geometric return, then ...


1

ok I found it 🙂and this works for any distribution, not just the normal distribution $f^*=\frac μ {σ^2 + μ^2} \approx \frac μ {σ^2} \space if μ\llσ$ here the steps: https://www.dropbox.com/s/4nqd5yfk2xcuag5/kelly.pdf


1

Book with counterpart SwedishAlphaBank, with whom you margin in SEK: USD RUB SEK SEKPnL 0 0 0 0 Buy 100 SEK worth of USD/RUB, meaning buy USD and sell RUB. 100 -100 0 0 With RUB interest rate at 0 (!), USD/RUB moves to 1.1, USD/SEK stays flat at 1 100 -100 0 9.09 Square up back to SEK on USD and RUB: 0 0 9....


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